2017/09/21 by Peter B. Shalen, Shalen, Peter B.
Mathematics · #57R18 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1709.07413
openalex publication_date 2017/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathfrak M be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that π1(\mathfrak M) contains no hyperbolic triangle group. We show that if the underlying manifold |\mathfrak M| is irreducible, and |\mathfrak M| is irreducible for every two-sheeted (orbifold) covering \widetilde\mathfrak M of \mathfrak M, and if \rm vol \mathfrak M≤1.72, then dim H1(\mathfrak M;\mathbb Z2)≤ 15. Furthermore, if \rm vol \mathfrak M≤1.22 then dim H1(\mathfrak M;\mathbb Z2)≤ 11, and if \rm vol \mathfrak M≤0.61 then dim H1(\mathfrak M;\mathbb Z2)≤ 7. The proof is an application of results that will be used in the sequel to this paper to obtain qualitatively similar results without the assumption of irreducibility of |\mathfrak M| and |\widetilde\mathfrak M|.